Nonemptiness of \(\mathcal {A}_1\)  can be checked in time that is linear in the size of \(\mathcal {A}_1\)  [1]}, so it is exponential in the size of \(\mathcal {A}_R\) . Again, the nonemptiness algorithm returns a finite-state transducer \(T_1\)  that induces a strategy mapping \(\Sigma _\mathcal {I}^+\)  to \(\Sigma _\mathcal {B}\) .
